aa r X i v : . [ qu a n t - ph ] M a r Identical two-particle interferometry in diffractiongratings
Pedro SanchoDelegaci´on de AEMET en Castilla y Le´onOri´on 1, 47014, Valladolid, Spain
Abstract
We study diffraction and interference of indistinguishable particles.We consider some examples where the wavefunctions and detection prob-abilities can be evaluated in an analytical way. The diffraction pattern ofa two-particle system shows notorious differences for the cases of bosons,fermions and distinguishable particles. In the example of near-field in-terferometry, the exchange effects for two-fermion systems lead to theexistence of planes at which the probability of double detection is null.We also discuss the relation with the approach to systems of identicalparticles based on correlation functions. In particular, we shall see thatthese functions reflect, as in noise interferometry, the underlying periodicstructure of the diffraction grating.
Exchange effects, associated with the peculiar behaviour of indistinguishableparticles, have been extensively studied in the literature. In recent times thesestudies have followed two principal lines. On the one hand, the experimentalmeasurement of correlation functions has corroborated the existence of bunchingand antibunching effects. On the other hand, some interference phenomena arestrongly dependent on the distinguishable or undistinguishable nature of theparticles involved. In this paper we shall be mainly concerned with this secondline of research.In the case of photons, following the seminal work of Hong-Ou-Mandel(HOM)[1], the effects of the statistics of the particles have been extensivelystudied for interferences originated by the interaction in a beam splitter. Someauthors have considered the extension of that approach to the case of massiveparticles. In [2] (and references therein) the behaviour of fermions in Mach-Zehnder interferometers has been analyzed. The proposal presented in [3, 4],is also interesting, where an electronic HOM-type interferometer is used to de-tect entanglement. In all the above arrangements the particles can only be1etected in the output arms of a beam splitter or interferometer. We analysein this paper whether exchange effects are also present in other interferometricarrangements with a continuous spatial distribution. The natural framework todiscuss this possibility is that of diffraction gratings, which generate continuousspatial patterns.At this point we must signal the existence of other works by Mandel’s groupwhere the spatial dependence of two-photon interferences has been analysed[5, 6]: two photons generated by down-conversion interfere at a beam splitterand detectors are placed in the two output arms. Moving the detectors relativeto the beam splitter, a spatial two-photon interference pattern is observed. Atvariance with our goal, these interferences are generated at a beam splitter andnot at a diffraction grating.As a first approximation to the problem, we consider in this paper somesimple examples of diffraction and interference by gratings which can be solvedanalytically. This way we can illustrate the main ideas involved in the problemwithout addressing other more technical questions. To be concrete, we shallanalyse two archetypical systems:(i)
Diffraction by a single slit of a Gaussian wave packet.
This is probablythe simplest system where the effects can be analysed. We shall evaluate theprobability detection patterns showing sharp differences between two-particledistributions of bosons, fermions and distinguishable particles.(ii)
Near-field interferometry with periodic gratings.
The best known exam-ple of this type of arrangement is the Talbot Lau interferometer [7, 8], whichhas been used in a number of interesting studies in matter wave interferometry[8, 9]. In this case we shall find a novel characteristic associated with the statis-tics of the system. For a pair of indistinguishable fermions, there are planeswith a null probability of double detection.As signaled before, in the other principal line of study of indistinguishablesystems the focus is on the correlation functions. This line originated in the sem-inal work of Hanbury Brown and Twiss (HBT) [10], in which they counted jointdetections (in two separate detectors) of photons from different chaotic sources.The correlations showed that the photons tend to arrive bunched in groups.This behavior can be understood within a classical framework if one introducesfluctuating phases. In contrast, the opposite trend observed for fermions hasno classical analogue. Then the only framework able to completely describebunching and antibunching is the quantum one, where these effects can be ex-plained in an unified way in terms of constructive and destructive interference.In the last years, the correlation functions of free boson [11, 12, 13] and fermions[14, 15, 13] have been experimentally obtained, corroborating the existence ofbunching and antibunching effects. It is natural to ask for the behaviour ofthe correlation functions in interferometry and to see if the presence of inter-ference effects can modify the usual picture. A well-suited technique for theposition-sensitive counting of particles when one is concerned with the spatial2ependence of the correlations is noise interferometry. This technique convertsspatial patterns into an interference signal. In particular, in Refs. [12, 15] theexistence of periodic quantum correlations was shown between density fluctu-ations in an expanding atom cloud when released from an optical trap. Thesespatial correlations reflect the underlying ordering in the lattice trap. Noiseinterferometry, is useful for identifying quantum phases of ultracold atoms inperiodic potentials. We shall evaluate the correlation functions in the framework(ii) showing that, as in noise interferometry, they reflect the periodic structureof the underlying arrangement.The plan of the paper is as follows. In Sect. 2 we present the basic equationsand derive some general properties of the interference of two identical particlesin a diffraction grating. The two analytically solvable models considered in thepaper are discussed in Sects. 3 (diffraction) and 4 (near-field interferometry).The last one is divided into two subsetions devoted, respectively, to single- andmulti-mode states. Section 5 deals with the behaviour of correlation functions indiffraction gratings. Finally, in Sect. 6, we recapitulate on the principal resultsof the paper and consider the possibility of testing them.
The arrangement we consider here consists in a system of two identical particlesarriving on a diffraction grating. After the diffraction grating we place detectorsmeasuring the interference pattern, which is obtained after many repetitions ofthe experiment.We denote by ψ k ( x , t ) and ψ p ( y , t ) the wavefunctions of the two particles,where x and y are the coordinates of the particles and k and p are their wavevec-tors or momenta. When the particles are in single-mode states, they refer tothe wavevectors or momenta of these modes. On the other hand, if we aredealing with multi-mode distributions, they represent the mean values of thesedistributions. When the particles are identical the usual product wavefunction ψ k ( x , t ) ψ p ( y , t ), must be replaced by1 √ ψ k ( x , t ) ψ p ( y , t ) ± ψ p ( x , t ) ψ k ( y , t )) (1)In the double sign expressions the upper one holds for bosons and the lower onefor fermions. The probabilities associated with this wavefuntion are P ( x , y , t ) = 12 | ψ k ( x , t ) | | ψ p ( y , t ) | + 12 | ψ p ( x , t ) | | ψ k ( y , t ) | ± Re ( ψ ∗ p ( y , t ) ψ ∗ k ( x , t ) ψ k ( y , t ) ψ p ( x , t )) (2)We have an interference term that is not present for distinguishable particles,according to the standard interpretation of exchange effects as interference ef-fects. From now on, in order to avoid any possible confusion by the use of the3nterference concept in two different ways, we restrict it to the effects induced bydiffraction gratings. The interference effects associated with the indistinguish-able character of the particles will be denoted as exchange effects.It must be remarked that the (anti)symmetrization procedure is only physi-cally required when there is a non-negligible overlapping between the two wave-functions. If not, the particles must be treated as distinguishable ones. Theimportant point is that once the overlapping has taken place, due to the lin-earity of the evolution equations, the (anti)symmetric form of the two-particlewavefunction persists at subsequent times.Several consequences directly emerge from the above expressions:(a) For bosons in the same state ( p = k ), the probability of simultaneousdetection of two bosons at the same point is P ( x , x , t ) = 2 | ψ p ( x , t ) | . The prob-ability is proportional to the fourth power of the wavefunction modulus. Thisbehaviour was described in models of double ionization by massive particles [16]and double absorption of massive particles [17]. At this point a word of cau-tion is in order if one tries to experimentally corroborate the above probabilitydistribution. The probability in Eq. (2) for different points, x = y , can beexperimentally measured using two detectors placed at x and y . In the case ofdouble (boson) detection at the same point x = y , we can only use one detector,which must be able to distinguish between single- and double-detection events.This type of detector is already available for photons [18, 19], but up to ourknowledge, not for massive particles.(b) For fermions the wavefunction of the complete system can have nodalpoints not present for factorizable ones. For example, if the individual wave-functions obey (up to a global phase) the relations ψ k ( X , t ) = ψ p ( X , t ) for a pairof points X and Y , these points are nodal ones of the two-particle wavefunction(not present in the factorizable case ψ k ( X , t ) ψ p ( Y , t ) unless X or Y are them-selves nodal points). This property will play a central role in the discussion ofthe detection distributions in near-field arrangements.(c) The set of points ( X , Y , t ) has another interesting property (for bosonsand fermions). The detection probabilities at these points show a maximaldeviation with respect to distinguishable ones. A good measure to quantifyhow much the detection rate is enhanced or diminished by the presence of theexchange effects is given by the ratio of the detection rates for idistinguishableparticles and for the same state without exchange effects. It is defined at eachpoint and at a given time as R ( x , y , t ) = P ( x , y , t ) P NE ( x , y , t ) (3)where P NE denotes the probability distribution without interchange effects. Inthe particular case of the set ( X , Y , t ) we obtain the maximum difference, R ( X , Y , t ) = (1 ± | ψ k ( X , t ) | | ψ k ( Y , t ) | | ψ k ( X , t ) | | ψ k ( Y , t ) | = 1 ± X and Y (at a given time t ) do not need tobe very close. The particles were close enough only at a previous time, leadingto the (anti)symmetrization of the wavefunction, which persists at subsequenttimes.In order to go beyond these general properties and to look for other ef-fects, we must consider particular arrangements and evaluate the form of thewavefunctions in each situation. We do it in the next two sections. We consider in this section the diffraction of a pair of indistinguishable particlesby a slit. The rigorous analysis of the problem leads to the numerical integrationof Fresnels functions [20, 21]. A simpler approach to the problem is, followingFeynman [20], to replace the slit by a Gaussian slit. With this approximationwe have an analytically solvable problem. The wave packet generated by thisslit with soft edges has also a Gaussian profile in the direction parallel to the slit.The movement in the perpendicular direction is assumed to be unaffected. Asusual in the treatment of the problem, we consider a two-dimensional problem,neglecting the vertical axis. Moreover, as the movement in the perpendicularaxis is almost unaffected, the relevant physics is contained in the parallel axisand the problem reduces to an one-dimensional one.The simplest way to study multimode states is to assume that the wavevec-tor distribution of each particle is a Gaussian one [21]. Using the notation of Ref.[21] the one-dimensional Gaussian distribution is f ( k ) = (4 π ) / σ − / exp( − ( k − k ) / σ ) with σ the width of the distribution and k its central value. Thewavefunction is obtained by superposing a set of planes waves with that distri-bution ((2 π ) − R dkf ( k ) exp( i ( kx − E k t/ ¯ h )) with E k = ¯ h k / m )[21]: ψ ( x, t ) = C ( t ) exp (cid:18) − σ ( x − vt ) + ik (2 x − vt ) + i ¯ hσ x t/mµ ( t ) (cid:19) (5)with v = ¯ hk /m , C ( t ) = π − / (cid:18) σ + i ¯ hσtm (cid:19) − / (6)and µ ( t ) = 2 (cid:18) h σ t m (cid:19) (7)Note the absence with respect to the formula in [21] of a multiplicative coefficient2 in the last term of the numerator in the exponential.5s required by our assumption of a Gaussian slit, the wavefunction has theform (up to some phase terms) of a spatial Gaussian packet with time-dependentwidth σ − + (¯ h t σ /m ).Using the expression of the wavefunction, it is immediate to evaluate theprobability distribution for two indistinguishable particles. We assume both par-ticles to be in states of the type (5), with the same width and central wavevectors k and p . The final result is P ( x, y, t ) / | C ( t ) | = 12 (cid:18) e A + e B ± e ( A + B ) / cos (cid:18) µ ( t ) ( x − y )( k − p ) (cid:19)(cid:19) (8)with A = − σ µ ( t ) (( x − v k t ) + ( y − v p t ) ) (9)and B given by a similar expression with the interchange of v k and v p .From the above expressions it is clear that the detection probabilities fora pair of indistinguishable particles show notorious differences with respect tothose of distinguishable ones. We illustrate this point numerically in Fig. 1,where we compare the probability of simultaneous detection for bosons, fermionsand distinguishable particles. In order to simplify the presentation, we fix oneof the points, x = x , and evaluate the detection probability for arbitrary y .We choose for x , at a fixed time t , the value x = v k t which maximizes thedetection probability in the first exponential. The probability in the case ofdistinguishable particles is P dis ( x, y, t ) / | C ( t ) | = e A ( x = x ) = exp( − σ ( y − v p t ) /µ ( t )). Taking ¯ ht/m = 0 . k = 1 = − p and a width of the distribution σ = 0 .
125 we have the following distribution:6 . . . . . y p Figure 1: In the vertical axis we represent, in arbitrary units, p = P ( x , y, t ) / | C ( t ) | . In the horizontal axis we have the coordinate y , also inarbitrary units, and the point 0 corresponding to y = v p t . The red, black andblue curves correspond, respectively, to bosons, distinguishable particles andfermions.Several consequences emerge directly from the figure. As expected, the dis-tribution in the distinguishable case retains a Gaussian form centred aroundthe point y = v p t . In contrast, for both fermions and bosons, the mathematicalform of the curves departs from the Gaussian profile showing the typical form ofan interference pattern. The maximum probability for bosons is approximatelyat the same position of the peak of the Gaussian for distinguishable particlesand doubles its value. At the same position we have a null-probability point forfermions.All the above development has been done in terms of the diffraction by aslit. However, these expressions also describe two particles that interact (theremust be a strong overlapping between them) at a given moment and then con-tinue a free evolution. We can measure the positions of the particles after theinteraction, which are given by Eq. (8). Note that in this case that expressionis exact, whereas for the description of the diffraction by the slit, it is only anapproximation. After considering an example of diffraction, we now move to a purely inter-ferometric arrangement, which illustrates other types of effects associated withthe interchange terms. In particular, we focus on near-field interference in theeikonal approximation [8], which can be tackled analytically (for the limitationsof the eikonal approximation see [22]). One of the best known examples of7ear-field arrangement is the Talbot Lau one [7], which has important applica-tions in matter wave interferometry [8, 9]. In order to emphasize on the mainphysical ideas of the problem without excessive mathematical technicalities, ina first step we shall only consider particles in single-mode states, showing laterthat the main result persists (under adequate restrictions) in the more realisticframework of multi-mode states.
We consider a plane wave ψ o = exp( ikz ) which incides on a grating placed inthe plane perpendicular to the z-axis. If the grating function is a periodic one,the transmission function is given by T ( x ) = X n A n exp(2 πinx/d ) (10)with n an integer, d the grating period and A n = | A n | exp( iξ n ) the coefficientof the term n . The wavefunction at a distance L behind the grating is [8]: ψ Lk ( x ) = e ikL X n A n exp (cid:18) πinxd (cid:19) exp (cid:18) − iπn Lλ k d (cid:19) (11)with λ k the wavelength of the particle and x the coordinate of the axis in whichwe measure the interferences.We consider now the normalization of this wavefunction. From the expres-sion | ψ Lk ( x ) | = X n,m A ∗ n A m exp (cid:18) πi ( m − n ) xd (cid:19) exp (cid:18) − iπL ( m − n ) λ k d (cid:19) (12)we see that R ∞−∞ | ψ Lk ( x ) | dx is not bound. In effect, the integration of theconstant terms A ∗ A , A ∗ A , · · · does not give a finite value. One possibil-ity to overcome this difficulty is to do the normalization in a relative sense, N − R ∞−∞ | ψ Lk ( x ) | dx = 1, with N = ( P n | A n | ) R ∞−∞ dx . An alternative form ofnormalization is to consider the integration over the period d , d R d dx | ψ Lk ( x ) | =1. Taking into account that R d dx exp(2 πinx/d ) = dδ n , we have d R d dx | ψ Lk ( x ) | = P n | A n | = A , and the normalization condition is obtained dividing by A . Itis simple to see that the same condition is reached using the relative normal-ization condition. The normalized wavefunction after the diffraction arrange-ment is given by ψ Lk ( x ) /A , or equivalently by Eq. (11) with the replacement A i → a i = A i /A ( a i = | A i /A | exp ( iξ n )).As anticipated in Sect. 2 let us consider the points X where the condition ψ Lk ( X ) = ψ Lp ( X ) holds (up to a global phase). In this example, the problem8s stationary and we do not need to include explicitly the time variable. Weintroduce the new parameter φ kp = πLd ( λ k − λ p ) (13)For all the initial wavevectors fulfilling the condition φ kp = 2 πn ∗ , with n ∗ aninteger, we have that at all the points X contained in the plane defined by z = L , the wavefunctions obey the relation ψ Lk ( X ) = e i ( k − p ) L ψ Lp ( X ) (14)From now on, we restrict our considerations to fermions. The antisymmetrizedwavefunction at that plane becomes null:1 √ ψ Lk ( X ) ψ Lp ( Y ) − ψ Lp ( X ) ψ Lk ( Y )) → √ e i ( k − p ) L ( ψ Lp ( X ) ψ Lp ( Y ) − ψ Lp ( X ) ψ Lp ( Y )) = 0 (15)For all the initial wavevectors for which the relation φ kp = 2 πn ∗ holds, we havethat in the plane z = L , there cannot be two-fermion detections. Conversely,once the values of λ k and λ p are fixed, there are always planes in which thephenomenon can be observed. These planes are given by z = L n , with L n =2 nd / ( λ k − λ p ) and n any integer. All the points in these planes are nodal pointsof the two-fermion wavefunction.Note that this result does not forbid one-fermion detections at these planes.In effect, the one-fermion detection probabilities are given by the reduced de-tection probabilities, P ( X ) = R P ( X, y ) dy . The integration on y contains any y placed on any plane, not only the Y ′ s contained in the nodal plains. Thecontributions of the points in the nodal plains are null, P ( X, Y ) = 0, but thecontributions P ( X, y ) with y outside these special planes can be different fromzero. The sum of all these contributions can be, in principle, different from zeroleading to a non-null one-fermion distribution at the nodal planes. We move now to the more realistic case of non-monochromatic states. We showthat the result of the previous subsection is still valid for narrow enough initialwavepackets.If the momentum distribution of the wavepacket is f ( k ), the wavefunctionat a distance L behind the grating becomes ψ Lf ( x ) = X n A n F n exp (cid:18) πinxd (cid:19) (16)9ith F n = Z dkf ( k ) e ikL e ( − iπn Lλ k /d ) ∼ e ik L e ( − iπn Lλ k /d ) Z dke − ( k − k ) / σ e i ( k − k ) L e ( − iπn L ( k − k ) /d kk ) (17)where we have adopted for the mode distribution a Gaussian curve with thecentral wavevector k . In order to simplify the notation, we have dropped thecoefficients associated with its normalization.The two imaginary exponentials inside the integral have a joint argument L ( k − k )(1 + ( πn /d kk )). When the condition d kk ≫ πn holds, theintegral simplifies to I ( k ) = R dke − ( k − k ) / σ e i ( k − k ) L . Let us consider thecircumstances under which this approximation can be justified. Three condi-tions must be fulfilled. (1) n must be small. For large values of n the right-handside of the inequality will be very large and the term containing it cannot beneglected. Thus, the summation in (16) must be truncated at some value n .This truncation is plainly justified because most of the behaviour of the systemcan be described taking into account only the lower terms in the summation. Asa matter of fact, in the representative experiment in [8] the interference signalis essentially determined by the 0 and ± k close to the central one, k , we must have d kk ≈ d k ≫ πn . This conditioncan be fulfilled by restricting our considerations to this range of central wavevec-tors. (3) As a consequence of (2), the distribution must be sharp enough around k in order the contribution of the elements with | k | ≪ | k | to be negligible.With this approximation Eq. (16) becomes: ψ Lk ( x ) = X | n |≤| n | A n I ( k ) e ik L e ( − iπn Lλ k /d ) e (2 πinx/d ) (18)Let us consider now another distribution with the same Gaussian form (the samewidth), but centered around another value p . Then I ( k ) = I ( p ), and we areat the same position of the previous subsection: when the relation φ k p =2 πn ∗ holds we have that in the plane z = L the wavefunctions obey ψ Lk ( X ) = e i ( k − p ) L ψ Lp ( X ).Within the range of validity of this approximation, the result obtained inthe previous subsection for a single-mode initial wavefunction can be extendedto its multi-mode counterpart. As signaled in the introduction, another fundamental line of research in identicalparticles is based on correlation functions. We analyse in this section whether10he presence of the interference device modifies the behaviour of the functionsfound in free systems or optical lattices.The correlation function is defined as C ( η ) = 1 d Z d P ( x, x + η ) dx (19)To calculate this expression we must evaluate the two-particle probability dis-tribution. In order to simplify the calculations, we restrict our considerations tothe case of particles described by single-mode plane waves. As we have seen inthe previous section, the result obtained for a mono-chromatic particle can beextended to sharp enough wave packets. We analyse the correlations in a fixedplane z = L . The two detectors measuring the correlations must be placed inthat plane. Then it is not necessary to include the L parameter explicitly as asuperscript of the wavefunction.The modulus of the initial state is: | ψ k ( x ) | = 1 + 2 m>n X n,m | a n a m | cos (cid:18) π ( m − n ) xd − πL ( m − n ) λ k d + ξ m − ξ n (cid:19) (20)In order to complete the expression for the probability, we must evaluate thecrossed or interference term Re ( ψ ∗ p ( y ) ψ ∗ k ( x ) ψ k ( y ) ψ p ( x )) = Re X n,m,r,s a ∗ n a ∗ m a r a s × exp (cid:18) πi [( s − n ) x + ( r − m ) y ] d (cid:19) exp (cid:18) − iπL [ λ p ( s − m ) + λ k ( r − n )] d (cid:19) = X n,m,r,s | a n a m a r a s | × (21)cos (cid:18) π [( s − n ) x + ( r − m ) y ] d − πL [ λ p ( s − m ) + λ k ( r − n )] d + ξ r,s,n,m (cid:19) where ξ r,s,n,m = ξ r + ξ s − ξ n − ξ m . Adding the direct and crossed terms, weobtain for the probability P ( x , y ) = 1 + m>n X m,n | a n a m | (cos ϕ km,n ( x ) + cos ϕ pm,n ( x ) + cos ϕ km,n ( y ) + cos ϕ pm,n ( y )) +2 m>n,r>s X n,m,r,s | a n a m a r a s | (cos ϕ km,n ( x ) cos ϕ pr,s ( y ) + cos ϕ pm,n ( x ) cos ϕ kr,s ( y )) ± X n,m,r,s | a n a m a r a s | × (22)11os (cid:18) π [( s − n ) x + ( r − m ) y ] d − πL [ λ p ( s − m ) + λ k ( r − n )] d + ξ r,s,n,m (cid:19) where ϕ km,n ( x ) = π ( m − n ) xd − πL ( m − n ) λ k d + ξ m − ξ n . Now, the correlationfunction can be obtained from Eq. (22). The calculation is simple but lengthy.There are four types of contributions: (i) constant terms, (ii) terms containingcos ϕ km,n ( x ), (iii) terms with cos ϕ km,n ( x ) cos ϕ ps,r ( y ) and (iv) those contained inthe crossed term. Let us evaluate them separately.(i) For constant terms we have, d R d ctedx = cte .(ii) By direct calculation we have, d R d cos ϕ km,n ( x ) dx = 0 = d R d cos ϕ km,n ( x + η ) dx . The terms with this form do not contribute to the correlation function.(iii) Using standard trigonometric relations and integrals we have that theintegral d R d cos ϕ km,n ( x ) cos ϕ pr,s ( x + η ) dx equals 0 when m − n = ± ( r − s ) and12 cos (cid:18) π ( s − r ) ηd + φ kpmnsr (cid:19) + 12 cos (cid:18) π ( s − r ) ηd + φ pkmnsr (cid:19) (23)otherwise, with φ kpmnsr = − πLd [( m − n ) λ k + ( r − s ) λ p ] + ξ m − ξ n − ξ s + ξ r .(iv) For the crossed term, we must distinguish when s − n + r − m is zero ordifferent from zero. When it is not zero, we have again the situation (ii) and itscontribution is null. On the other hand, when s − n + r − m = 0, the integralof the cosine gives cos (cid:18) π ( r − m ) ηd + Θ kpnmrs (cid:19) (24)with Θ kpnmrs = − πLd [( r − n ) λ k + ( s − m ) λ p ] + ξ r,s,n,m .Combining all these expressions, we obtain the correlation function: C ( η ) = 1 + 2 m>n,s>r X n,m,r,s | a m a n a s a r | × (cid:18)
12 cos (cid:18) π ( s − r ) ηd + φ kpmnsr (cid:19) + 12 cos (cid:18) π ( s − r ) ηd + φ pkmnsr (cid:19)(cid:19) ± s − n + r − m =0 X n,m,r,s | a m a n a s a r | cos (cid:18) π ( r − m ) ηd + Θ kpnmrs (cid:19) (25)The correlation function depends on the parameters of the grating, a i and d and on the wavevectors of the incident particles (through φ kpmnsr and Θ kpnmrs both functions of λ k and λ p ). This behaviour is to be compared with that offree particles, for which the correlation function depends on the wavevectors inthe form C free ( η ) = 1 ± cos(( p − k ) η ) (for plane waves exp( ikx ) and exp( ipy )).The spatial periodicity of C free is determined by p − k . For interfering particles,however, the periodicity is only ruled by d/n (with n any integer), a functionof the parameter of the grating d . In the correlation function we have all the12eriods ( d/n ) contained in the diffraction grating. The wavevectors only deter-mine (in conjunction with d and the coefficients ξ i ) a phase for each term. Weconclude that, as in noise interferometry, the correlation functions reflect theunderlying structure of the periodic grating.A particularly clear illustration of the behaviour of the correlation functionsis obtained in the particular case that only a and a are different from zero.Moreover, for the sake of simplicity, we assume the two coefficients to be real.The correlation function in this case is (using a + a = 1) C ( η ) = (1 ± ± a a ( − φ kp ) + 2 a a ( ± φ kp ) cos (cid:18) πηd (cid:19) (26)Now there is only one periodicity, d . The wavevectors only determine (in con-junction with a o and a ) the amplitude of the oscillations. Moreover, the de-pendence is in the form k − p , instead of p − k .For short distances, η →
0, we have C F ( η ) ∼ C ( πd ) η ( C = 4 a a (1 − cos φ kp ))and C B ( η ) − C B (0) ∼ − a a ( πd ) (1 + cos φ kp ) η that is the usualdependence on η , but with different coefficients. In this paper we have analysed the behaviour of exchange effects in diffractionand interference. We have found the existence of some distinctive characteristicsof this type of interferometry: modifications of the probability distributionswith respect to those of distinguishable particles, strong increase or decreaseof the double detection rates for some pairs of points (similar to bunching orantibunching, but without need of closeness between the particles) and existenceof planes where the double detection of fermions is forbidden. On the otherhand, the correlation functions show a behaviour similar to that found in noiseinterferometry, reflecting the periodic structure of the diffraction grating.Now briefly address the question of how the above distinctive characteristicsof continuous interferometry of identical particles can be tested experimentally.As signaled before, the interchange effects are present (and persist in the sub-sequent evolution of the system) when there is a non-negligible overlappingbetween the wavefunctions of the two particles at a given time. Then the keystep is to obtain a non-negligible overlapping of the two particles at the timethey reach the diffraction grating. This is equivalent to a careful preparationof the times of arrival of the two particles. In other words, the peaks of thetwo distributions must reach the grating at the same time. Because of recentprogress in matter wave interferometry, interferometry of heavy molecules andthe existence of massive systems with high degrees of coherence, these effectsseem to be accessible to experimental scrutiny. Conversely, the presence of theeffects described in this paper could be used as a test of the closeness of twoparticles under some specific preparation.13t must also be noted that because of the interaction of the beam with thegrating, there can be absorption and backscatter processes which result in caseswith zero or one particles arriving at the detectors. We can overcome thisdifficulty by considering a postselection process in which only events with twodetections are taken into account.In this paper we have restricted our considerations to two simple examplesthat can be treated analytically. Other types of diffraction gratings must bestudied in order to see if (the same or different) exchange effects are also present.
Acknowledgments
We acknowledge partial support from MEC (CGL 2007-60797).
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